“There is an astonishing imagination, even in the science of mathematics. An inventor must begin with painting correctly in his mind the figure, the machine invented by him, and its properties or effects. We repeat there was far more imagination in the head of Archimedes than in that of Homer.” — Voltaire

Brood War

A newlywed couple are planning their family. They’d like to have four children, a mix of girls and boys. Which is more likely: (1) two girls and two boys or (2) three children of one sex and one of the other? (Assume that each birth has an equal chance of being a boy or a girl.)

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The Christmas Truce


As Christmas approached in 1914, a number of impromptu cease-fires broke out on the Western Front in which German and British troops exchanged greetings, song, and even food. Rifleman Oswald Tilley of the London Rifle Brigade wrote to his parents on Dec. 27 regarding an incident near Ploegsteert, just north of the Franco-Belgian border:

On Christmas morning as we had practically ceased to fire at them, one of them started beckoning to us so one of our Tommies went out in front of our trenches and met him halfway amidst cheering. After a bit a few of our chaps went out to meet theirs until literally hundreds of each side were out in No Man’s Land shaking hands and exchanging cigarettes, chocolate and tobacco etc. … Just you think that while you were eating your turkey etc. I was out talking and shaking hands with the very men I had been trying to kill a few hours before. It was astonishing!

In subsequent years the authorities tried to discourage such truces. Apart from reproving the breakdown in discipline, they had trouble getting the war started again. In late 1915 Ethel Cooper, an Australian woman living in Germany, met a soldier home on leave from the XIX Saxon Corps who told her that his unit had fraternized extensively with a British battalion for two days beginning that Christmas Eve. She wrote, “The trouble began on the 26th, when the order to fire was given, for the men struck. Herr Lange says that in the accumulated years he had never heard such language as the officers indulged in, while they stormed up and down, and got, as the only result, the answer: ‘We can’t — they are good fellows, and we can’t.’ Finally, the officers turned on the men, ‘Fire, or we do — and not at the enemy.’ Not a shot had come from the other side, but at last they fired, and an answering fire came back, but not a man fell. ‘We spent that day and the next day,’ said Herr Lange, ‘wasting ammunition in trying to shoot the stars down from the sky.'”

(From Marc Ferro et al., Meetings in No Man’s Land, 2007)


In 1981, when science journalist Marcus Chown was an undergraduate physics student, his mother watched a profile of Richard Feynman on the BBC series Horizon. She had never shown an interest in science before, and he wanted to encourage her, so when he advanced to Caltech to study astrophysics, he told Feynman of his mother’s interest and asked him to send her a birthday note. She received this:

Happy Birthday Mrs. Chown!

Tell your son to stop trying to fill your head with science — for to fill your heart with love is enough!

Richard P. Feynman (the man you watched on BBC “Horizons”)

The Centipede Game

Before you are two piles of coins. One contains 4 coins and the other contains 1. If you like, you can keep the larger pile, give me the smaller, and end the game. Or you can pass both piles to me. In that case the size of each pile doubles and I’m given the same option — I can keep the larger pile and give you the smaller one, or I can pass both piles back to you, in which case they’ll double again.

We both know that the game will end after six rounds. At that point I’ll have the coins and will win 128 coins to your 32. You’d be better off stopping the game in round 5, when you’ll have 64 coins and I have 16. But, by similar reasoning, I’d prefer round 4 to round 5, and you’d prefer round 3 to round 4 … if we rely on each other to be purely rational, it seems your best opening move is to end the game at once and keep 4 coins. This is less than you’d make in round 6, but it appears that purely rational play will never reach that round.

In practice, interestingly, human beings don’t do this — almost no one stops at the first opportunity, even after several repetitions of the game. Why they do so is not clear — possibly they’re hoping that their opponent has not reasoned through the whole game, or perhaps they’re agreeing tacitly to cultivate the pot in hopes of being the first one to cash out abruptly; perhaps the satisfaction of anticipating such a victory makes the risk worthwhile.

In lab tests in 2009, economists Ignacio Palacios-Huerta and Oscar Volij found that only 3 percent of games between students ended in the first round, but 69 percent of games between chess players did so. This rose to 100 percent when the first player was a grandmaster. They conclude that the most important factor is common knowledge of the players’ rationality, rather than altruism or social preferences.

(Ignacio Palacios-Huerta and Oscar Volij, “Field Centipedes,” American Economic Review 99 (4): 1619–1635.)

Party Planning

An Englishman buys a horse and hires porters to take the horse up to his apartment on the fourth floor. The porters exert themselves and sweat. Finally they succeed in getting the horse to his apartment.

He asks them to put the horse in the bathtub.

After they finish the job, one of the porters asks him, “Why do you need a horse in the bathtub?”

The Englishman says, “Well, tomorrow evening I’m having a party at home. One of the guests will go into the bathroom, see the horse, come to me and say, ‘You know you have a horse in your bathtub.’ And I’ll tell him, ‘So what?'”

— Sion Rubi, Intelligent Jokes, 2004

The Defenestrations of Prague


On July 30, 1419, Czech priest Jan Želivský was leading his congregation through the streets of Prague to protest corruption in the Catholic church when someone threw a stone at him from the window of the town hall. His followers stormed the hall and threw 13 members of the town council from a high window, killing them.

Remarkably, the same thing happened again in 1618, when King Ferdinand dissolved the Protestant estates in Bohemia. Aggrieved Protestants confronted Catholic officials in the chancellory and threw several of them from a third-floor window. All three survived — Catholics contended that they had been saved by angels, Protestants that they had landed on a dunghill. (Or, a reader suggests, “the Czechs bounced.”)

Black and White

meredith chess problem

By William Meredith. White to mate in two moves.

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Flip Sum

A problem from the 1999 St. Petersburg City Mathematical Olympiad:

Fifty cards are arranged on a table so that only the uppermost side of each card is visible. Each card bears two numbers, one on each side. The numbers range from 1 to 100, and each number appears exactly once. Vasya must choose any number of cards and flip them over, and then add up the 50 numbers now on top. What’s the highest sum he can be sure to reach?

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Out of This World

Tunisia has an unique tourist draw — its southern desert contains the abandoned sets of five Star Wars movies, in which the Sahara stood in for the planet Tatooine.

A list, complete with photos and geographical coordinates, is here.

They’re popular with European tourists, but they won’t last — the 20-building set of Mos Espa, Anakin Skywalker’s hometown, is being engulfed by sand dunes.